Practicing Unit conversions

...do we need to know how to convert units?

To make a comparison between two numbers, they must be in the same units!

And comparisons are useful...

  • to make a decision: Which one to buy?
  • to prioritize: Will I save more CO2 emissions by reducing my beef consumption, or buying fruit that's grown locally?
  • to make an argument: The city of Goshen would come out ahead by paying for curbside recycling to reduce the money they spend to send all of our garbage to the landfill!
  • to communicate urgency (or lack of urgency) to an audience:
  • to check your answers to a math problem: Wait a second, I'm supposed to calculate the size of an atom, but when I punched the numbers into my calculator, I got 10 feet?!

The Inflation Reduction Act of 2022 "makes the single largest investment in climate and energy in American history". Along with \$300 billion in deficit reduction, it will disburse \$369 billion dollars to address energy security and climate change over the next 10 years, or \$37 billion per year in energy/climate funding.

What should we make of this comparison with major oil company profits in 2022??

Standard metric prefixes:

One kilogram is 1,000 grams; One kilometer is 1,000 meters; One kilowatt is 1,000 watts.

These are used so often, and they're very regular, and they are tacked onto all different *kinds* of units. I'll expect you to memorize these, if you don't know them already:

  • kilo- =1,000 = $10^3$ = one thousand
  • mega- = 1,000,000 = $10^6$ = one million : 1 followed by 6 zeroes.
  • giga- = 1,000,000,000 = $10^9$ = one billion.

  • milli- =1/1,000=0.001=$10^{-3}$: Starting with 1.0, move the decimal point 3 places to the left.
  • micro- =1/1,000,000=0.000 001=$10^{-6}$
  • centi- = 1/100=0.010=$10^{-2}$

English (American) units

On a test or quiz, I'll give you these. Don't bother to memorize:

  • 1 mile = 5,280 feet = 1.6 kilometers
  • 1 inch = 2.54 cm    
  • 12 inches = 1 foot, OK, you probably do know this one!
  • 1 kg = 2.2 lbs; 16 oz = 1 lb

21st century: Google!

Google can do many unit conversions for you:

Google will respond with useful information to queries like 1 US dollar in Icelandic Krona or 170 lbs to kilograms.

This sign as you enter Österreich(= Austria) indicates that the default speed limit on high speed two-lane roads is 100 kilometers / hour, on freeways 130 km / hr, and in towns is 50 km / hr. Find out how fast you can drive on a freeway using 130 km/hr to miles/hr...

Practice unit conversions

But you need to know how to make these unit conversions yourself: Many conversions--particularly between metric units--can be done without finding the nearest Internet portal. And you can also use these techniques to solve more complicated questions (such as those on the combustion worksheet...) or that you can't quite so easily ask Google.

Directions

  • In what follows, show your work for each unit conversion.
  • Every number you write down should have units written down with it.
  • Answers without your work will receive no credit, because we know you can just Google the answer!

But you may use Google to check your answers and (and then correct your work, as necessary). For example: How tall (in meters) is a 6 ft tall person?

Start with: $$ 6 \text{ feet} \cdot \frac{?}{? \text{ feet}} \cdot \text{[? more conversions]}= \text{[ ? ] meters}.$$

$$\begineq 6 \text{ ft} \cdot \frac{12 \text{ in}}{1 \text{ ft}}\cdot \frac{2.54 \text{ cm}}{1 \text{ in}}\cdot \frac{1 \text{ m}}{100 \text{ cm}}&= \frac{6\cdot 12 \cdot 2.54}{100}\text{ m}\\ &={\color{blue}1.83 \text{ m}}\endeq$$

A common error!

When there are multiple numbers in the denominator: E.g. convert 10,000 seconds into hours: $$ 10,000 \text{ sec }\cdot \frac{1 \text{ minute}}{60 \text{ seconds}}\cdot \frac{1 \text{ hour}}{60 \text{minutes}}=\frac{10,000}{60*60}\text{ hours}.$$

It's tempting to punch this into your calculator as:
10000 / 60 * 60 means $\frac{10000}{60}*60$= 10,000??!!

Instead enter this as:
10000 / 60 / 60 $\approx$ 2.8 hours (a little less than 3 hours).

 

  1. The distance from Goshen to Elkhart is 11.3 miles. Convert to kilometers.

    $$11.3 \text{ miles} *\frac{1.6 \text{ km}}{1 \text{ mile}}= 18.1 \text{ km}$$




  2. The distance from Chicago to Goshen is 196 km. Convert that to miles.

    $$196 \text{ km} *\frac{1 \text{ mile}}{1.6 \text{ km}}= 122.5 \text{ miles}$$




  3. You're speeding along the highway at 75 miles/hour. How fast is that in kilometers / hour?

    Write 75 mph as the fraction "75 miles / 1 hour". We want "kilometer / 1 hour". So you can leave the hours alone, and just convert the 75n miles to kilometers... $$\frac{75 \text{ miles}}{1 \text{ hr}} \frac{1.6 \text{ km}}{1 \text{ mi}}= \frac{120 \text{ km}}{1 \text{ hr}} $$




  4. How fast is 75 miles/hour in meters/second?

    $$ \frac{120 \text{ km}}{1 \text{hr}} \frac{1000 \text{ m}}{1 \text{ km}} \frac{1 \text{ hr}}{60 \text{ min}} \frac{1 \text{ min}}{60 \text{ sec}}= \frac{33 \text{ m}}{1 \text{ sec}} $$






  5. 200 lbs is how many kilograms?

    $$200 \text{ lbs} *\frac{1 \text{ kg}}{2.2 \text{ lbs}}=91 \text{ lbs}$$




  6. 14 grams is how many kilograms?

    $$14 \text{ gm} *\frac{1 \text{ kg}}{1000 \text{ gm}}= 0.014 \text{ kg}$$




  7. A heavyish letter weighs 1 ounce (oz). How much is this in grams?

    $$1 \text{ oz} *\frac{1 \text{ lb}}{16 \text{ oz}} *\frac{1 \text{ kg}}{2.2 \text{ lb}} *\frac{1000 \text{ g}}{1 \text{ kg}} = 28 \text{ g}$$




  8. Convert the thickness of the soap film you found, which was in millimeters, in the soap/pepper demo into meters. Compare your answer to some atoms or molecules in the Scale of the Universe. What objects might be "reasonably close"??

    You each had slightly different values for the thickness, which is to be expected. I'll take one value, for example, of $2.2\times 10^{-6}$ millimeters. Converting to meters... $$\begineq 2.2\times 10^{-6}\text{ mm }\cdot \frac{1 \text{ meter}}{1000 \text{ mm}} &=2.2\times\frac{10^{-6}}{1000}=2.2\times \frac{10^{-6}}{10^3}\\ &=2.2\times 10^{-6-3}=\color{blue}2.2\times 10^{-9}\text{ meters} \endeq $$

    $2.2\times 10^{-9}$ is the same as 2.2 nanometers. But on the scale of the universe, assuming the width of that oval is $10^{-9}$ meters...

    It looks like twice that width would be larger than a water or cesium atom--maybe 7 to 10 times the size of a water molecule?--but smaller than a phospholipid molecule. A bit larger than a glucose molecule...
    Googling "Soap molecule" I find this drawing of soap molecules (the gray cylinders) on top of water:
    chemistry.elmhurst.edu
    The soap molecules are approximately 5-7 times the size of the little red/white water molecules? So this estimate of $2.2\times 10^{-9}$ meters is pretty darn close to the size of soap molecules depicted in the diagram!